I got two fish today that were in a container that holds three so they share the same water. The sick one was laying on his side and breathing heavily, though he perked up slightly when the other two were taken away. I'm not sure what is wrong with him but I need an idea on how long it will take for me to know if the two I bought have caught it.

Also, one of the fish looks as if he's in shock. He's doing a lot of sitting on the bottom and breathing heavily. Any ideas of what I can do to make him more comfortable?

Housing
What size is your tank? 2.5 gallons (the healthy looking fish is in 2 gallons)
What temperature is your tank? 72
Does your tank have a filter? No
Does your tank have an air stone or other type of aeration? No
Is your tank heated? No
What tank mates does your betta fish live with? None

As a note, I plan to get heaters and filters, I just
came across these guys today and didn't have plans on getting a new one, let alone two.

Food
What type of food do you feed your betta fish? Flake food that doesn't have veggie ingredients.
How often do you feed your betta fish? Twice daily. I didn't feed the one in shock as I don't think he would have been up to eating.

Maintenance
How often do you perform a water change? A couple of times a week
What percentage of the water do you change when you perform a water change? 50% but I do full changes when necessary
What type of additives do you add to the water when you perform a water change? Water conditioner

Water Parameters:
Have you tested your water? If so, what are the following parameters? Not tested

Ammonia:
Nitrite:
Nitrate:
pH:
Hardness:
Alkalinity:

Symptoms and Treatment
How has your betta fish's appearance changed? He looked lively at the store but now he sits still at the bottom of the tank for the most part
How has your betta fish's behavior changed?
When did you start noticing the symptoms?
Have you started treating your fish? If so, how?
Does your fish have any history of being ill?
How old is your fish (approximately)? I'm not really certain how to guess the age, but the one who isn't doing good is definitely grown up; the one who is doing well so far looks young, though he has his full fins.

I wouldn't filter anything smaller than a 5g because it just creates current the fish can't get away from it. Also trying to cycle anything smaller will require more careful monitoring and more supplies and it probably still won't ever fully establish.

If you did want to do that I would use a sponge filter, with pump and airline. Then you will need a drops kit for ammonia, nitrite and nitrate. You need to be testing daily with a reliable drops kit for ammonia and nitrite and doing an extra 50% change any time you see either. In addition to this a weekly 50% with siphon or new fresh turkey baster that has never seen chems is needed to remove poop and other debris from the gravel. It is not enough to just scoop water off the top ever.

First you will see ammonia, then nitrite. Eventually, hopefully, you will see ammonia fall and stay at 0 even after a week of no water changes, and finally nitrite. At this point you will be left with only nitrates after a full week of no changes and these can be kept <20ppm by twice weekly 50% change with baster/siphon. However, cycling will take up to two months to complete and many if not most people will tell you that you can't cycle a tank of this size and you will always need twice weekly 50% and 100% water changes or you will always see ammonia continuing to build and the cycle will never finish..

If you don't cycle it and while you are not cycling it you should do two water changes a week of 50% and 100%.

The 50% changes the betta can be kept in the bowl and use a turkey baster to remove half the water and as much of the debris as possible. For the 100% you need to remove him - scoop him out with a plastic solo type cup and set aside while you thoroughly rinse the bowl and gravel to remove the debris. Then he should be acclimated to the new water by floating for an hour while you slowly add a couple tablespoons of new water to the cup every 10 minutes. When you release him, try to let as little of the old cup water back into the tank as possible. All water changes should use same temp water, matched to running tap using the in tank thermometer and the water needs to be premixed with conditioner before adding it to the betta tank. If you don't already have anything, you can use gallon water jugs from the grocery store - rinsed thoroughly in hot water but no chems.

I always use the siphon. I've never checked ammonia, but these two new fish just got home this afternoon so I haven't touched their water. When I set the tanks up I made sure the water was the same temperature as what was in the bags they came in and I put the stuff in to neutralize the tap water.

You'll just have to give them time and hopefully they will recover. Try to reduce stress as much as possible.. keep the room dark and quiet and maybe even cover it with a towel

In the future, you should transfer the bagged betta to a plastic solo type cup and float him in the fully heated new tank for at least an hour while you very slowly add a couple tablespoons of new water to the cup every 10 minutes. I also suggest specifically using Prime in the new water at a double dose and you can even do this acclimation longer for 2 or 3 hours to be really safe.

Thank you. I never knew they were supposed to be changed so slowly. I've never had this problem before but I'll make sure to do that in the future. I'm glad you mentioned keeping him in the dark as I've had the light on thinking it would make him feel more comfortable not being in the dark.

I forgot all about adding photos. Here's the one I'm really worried about. For some reason it won't let me put more than one photo and for this I apologize. The first shows that at times he does look normal.